Meta

Tag: Interpolation

Most of my Linux-computers have as their sound-server ““. But specifically on my laptop named ‘‘, I have set up the to be able to run as an alternative, yet not to be running by default. I have performed experiments on that laptop, to confirm that I can launch this sound-server, using a GUI named ‘‘, but have also had to make modifications to how this GUI executes commands from the user, so that its start-up pauses the , which has been able to resume successfully after I was done using . Without such a detail, the attempt should not be made.

The reason this is possible is the fact that individual sound devices are just clients to that daemon, including any number of devices that act as sound-sources, rather than acting as sinks, i.e. that act as inputs rather than as one output.

I also own a USB-Sound-Device named the ‘‘, which is mainly intended for use in sound capture, but which also has outputs intended for monitoring purposes.

If I was to run at 192kHz, then one simple consequence of that would be, that zero actual sound-devices would remain compatible with it. As to how cleanly an attempt to connect to an incompatible device exits, giving error messages or crashes, I have not tested, because when I tested the , I took into account the real limit of that device at 96kHz.

Similarly, the runs with 32-bit linear precision by default. In this case, when we enable devices to act as clients, which are only capable of 24-bit sample-depth, which is common, the mismatch is safely ignored. already sees to it, that the last 8 bits of precision get ignored.

Now, I could be cautious and worry, that because of errors in the Linux drivers, those last 8 bits somehow get mapped to a control register as an error. But then the simple way to test for that, was simply to send some 32-bit sound through JACK, to this output device. What I found when testing this, is that the basic operation of the was not disturbed, even though my hearing was not good enough, to tell me when I had my Sennheisers on, whether in fact 24-bit precision was still working. I was mainly testing, that trying to send a 32-bit value, does not disrupt the actual operation.

In This Posting, I elaborated at length, about Polynomial Approximation that is not overdetermined, but rather exactly defined, by a set of unknown (y) values along a set of known time-coordinates (x). Just to summarize, If the sample-time-points are known to be arbitrary X-coordinates 0, 1, 2 and 3, then the matrix (X1) can state the powers of these coordinates, and If additionally the vector (A) stated the coefficients of a polynomial, then the product ( X1 * A ) would also produce the four y-values as vector (Y).

X1 can be computed before the algorithm is designed, and its inverse, ( X1^-1 ), would be such that ( X1^-1 * Y = A ). Hence, given a prepared matrix, a linear multiplication can derive a set of coefficients easily from a set of variable y-values.

Well this idea can be driven further. There could be another arbitrary set of x-coordinates 1.0, 1.25, 1.5, 1.75 , which are meant to be a 4-point interpolation within the first set. And then another matrix could be prepared before the algorithm is committed, called (X2), which states the powers of this sequence. And then ( X2 * A = Y' ), where (Y') is a set of interpolated samples.

What follows from this is that ( X2 * X1^-1 * Y = Y' ). But wait a moment. Before the algorithm is ever burned onto a chip, the matrix ( X2 * X1^-1 ) can be computed by Human programmers. We could call that our constant matrix (X3).

So a really cheap interpolation scheme could start with a vector of 4 samples (Y), and derive the 4 interpolated samples (Y') just by doing one matrix-multiplication ( X3 * Y = Y' ). It would just happen that

Y'[1] = Y[2]

And so we could already guess off the top of our heads, that the first row of X3 should be equal to ( 0, 1, 0, 0 ).

While this idea would certainly be considered obsolete by standards today, it would correspond roughly to the amount of computational power a single digital chip would have had in real-time, in the late 1980s… ?

I suppose that an important question to ask would be, ‘Aside from just stating that this interpolation smooths the curve, what else does it cause?’ And my answer would be, that Although it Allows for (undesirable) Aliasing of Frequencies to occur during playback, when the encoded ones are close to the Nyquist Frequency, If the encoded Frequencies are about 1/2 that or lower, Very Little Aliasing will still take place. And so, over most of the audible spectrum, this will still act as a kind of low-pass filter, although over-sampling has taken place.